Four Weights

When using a balance scale, weights can be placed on either side of the scale. For example, if a 10 pound weight provides a counter balance to an object and a 7 pound weight, then the object must weigh 3 pounds. What four weights can be used to weigh objects of 1, 2, 3 ... 38, 39, 40 pounds?

Hi,

There are a number of approaches running from trial and error to use of recurrences and the binomial theorem but the simplest way to look at this problem in general, i.e. to show how one can use n weights to balance the weights 1, 2, ... ,(3ⁿ - 1)/2, is to use what we call base 3 arithmetic. Without being too theoretical we will try to show you the algorithm of how to make a weight balance.
   Recall that numbers that we commonly use such as 3456 are what we call base 10 numbers, i.e.

   Suppose that you want to balance say thirty-one pounds. Imagine that you have put thirty-one pounds on each side of the balance. On the left is thirty-one pounds written in base 10 notation (31) and on the right is twelve pounds written in base 3 notation (1011₃). Hence on the right is 1*3³ + 1*3¹ + 1 so you can balance 31 pounds with one 27 pound weight, one 3 pound weight and one 1 pound weight. In a similar fashion any weight which has only ones and zeros when written in base 3 notation is easy to balance. For example, looking at the table above, 13 pounds balances with one 9 pound weight, one 3 pound weight and one 1 pound weight.
   Suppose now that you want to weigh thirty-three pounds. Thirty-three in base 3 notation is 1020₃ but you don't have two 3 pound weights to put on the right. Put your 3 pound weight with the thirty-three pound weight on the left side of the balance and then to balance you need an extra three pounds on the right. That is on the right you need one 27 pound weight and three 3 pound weights which is equivalent to one 27 pound weight and one 9 pound weight. That is 33 + 3 balances with 1020₃ + 10₃ = 1100₃. Thus put your 1 pound weight on the left with the 33 pound weight and balance it with a 27 pound weight and a 9 pound weight.
   As a second example look at seventeen pounds. Seventeen in base 3 is 122₃. Since there is a 2 in the first position (reading from the right) put your 1 pound weight on the left, then to balance on the right you need eighteen pounds. Eighteen is 200₃ which is two nines so again you need to put your 9 pound weight on the left, forcing another 9 pounds on the right for a sum of twenty-seven pounds. Thus 17 + 1 + 9 balances with 122₃ + 1₃ + 100₃ = 1000₃ which is 27 pounds. Thus to weight 17 pounds put the 1 and 9 pound weights on the left with the 17 pound weight and the 27 pound weight on the right.
   What you are actually doing here is looking for a number which contains only zeros and ones when written in base 3, that when added to 122₃ in base 3 results in a total that contains only zeros and ones. This number is 101₃ since

Cheers,
Penny